System and method for probabilistic WLAN positioning

ABSTRACT

This disclosure is directed to a wireless network node having position determination capabilities. The position of a node is determined using range measurements to other network nodes having known locations. Probability density functions modeling uncertainty factors are incorporated in the estimation algorithms to account for the dynamic nature of wireless network, including the relative motion of a node in the network. These probabilistic estimation techniques provide a solution in the form of an expectation value for the position of the network node and a variance that can be assessed to determine the validity of the position determination.

FIELD OF THE PRESENT INVENTION

This disclosure generally relates to positioning systems and morespecifically to systems and methods for providing a wireless networkcapable of making positioning determinations for nodes on the basis ofprobability densities.

BACKGROUND OF THE INVENTION

Widespread adoption of wireless communication standards has led to aproliferation of networks. Notable examples include systems based on theIEEE 802.11 protocols. With this growth, there has been a concurrentincrease in the functionality provided by such networks. A particularlydesirable feature is the ability to make positioning determinations fornodes within the network by analyzing the transmissions between thenodes. The benefits associated with such a capability are numerous.

One important application relates to the tracking of assets or people.By providing an entity with a transceiver in communication with thewireless network, its position can be determined rapidly. With regard toassets, knowledge of current position facilitates utilization of theasset, streamlines logistics and helps prevent loss. With regard topersons, management and oversight can be significantly improved.

Another application currently receiving considerable attention isLocation-Based Services (LBS). With the availability of positioninformation for mobile clients, the user of such a device can receiveinformation specifically tailored to the user's current location. Aswill be appreciated, the benefits associated with this feature extend tomultiple situations, including recreation, commercial and workenvironments.

In view of the desirability of providing network nodes having positiondetermination capabilities, a number of strategies have been employed toperform the necessary determinations. Many of the location techniquesused in wireless networks are based on electromagnetic analyses of thecharacteristics of communications between nodes of the network. Forexample, received signal strength indication (RSSI), time-of-arrival(TOA), time-difference-of-arrival (TDOA) and round trip time (RTT) canall be used to make estimations of the distance between participatingnodes. With sufficient numbers of distance estimations between a nodewith unknown position and nodes having known position, range-basedlocation determinations such as trilateration are possible.Alternatively, angle-of arrival (AOA) can also be used with suitabletriangulation algorithms to make position determinations, but suchtechniques typically require use of an antenna array to obtain thenecessary signal angle information.

To obtain the benefits of making position determinations for networknodes in a wireless network, a number of challenges must be met. In oneimportant aspect, the distance estimations used for range-basedsolutions require are subject to uncertainties resulting from many ofthe same noise and interference issues that effect wirelesscommunication in general. For example, multipath distortion affectingthe wireless channel may lead to distance estimates that do notcorrespond to the straight line path between the nodes. As will beappreciated, multipath propagation errors are typically exacerbated inindoor environments due to the many reflecting surfaces.

Accordingly, position determination in a wireless network environmentrequires robust strategies for managing noise and interference so thatuseful estimations can be made. Conventional methods for dealing withnoise and interference include the use of data fitting algorithms suchas a least squares-based analysis. Under these approaches, positiondeterminations can often be improved by mitigating noise using filteringtechniques including the use of Kalman filters and the like. In general,these methods provide relatively good position determinations whenample, high quality measurements are available.

The above noted conventional approaches are less successful, however,when only more sporadic measurements are available. In many practicalsituations, then, there may be an insufficient number of rangemeasurements to obtain reliable position determinations. Further,relative motion between the network nodes also significantly underminesthe ability of conventional techniques to provide sufficiently accurateposition determinations. In addition to the noise imparted by motion,movement of one or more nodes may also render older measurementsunusable.

As a result, it would be desirable to provide position determination forwireless networks even when relatively few measurements are available.Moreover, it would be desirable to provide such position determinationusing measurements taken at different times, even when relative motionbetween nodes may occur. Yet further, it would be desirable to providesuch position determination so that the quality of the determination maybe easily assessed. It would also be desirable to reduce the filteringnecessary to obtain position determinations. This invention accomplishesthese and other goals.

SUMMARY OF THE INVENTION

In accordance with the above needs and those that will be mentioned andwill become apparent below, this disclosure is directed to a method formaking a position determination for a first node within a wirelessnetwork including by obtaining position information for a plurality ofknown network nodes, obtaining range measurements for distances betweenthe first node and the plurality of known network nodes, determining asingle range measurement probability density function for a position ofthe first network node for each of the range measurements, determining acombined probability density function by taking the product of the singerange measurement probability density functions, and determining anexpectation value corresponding to the first network node's position anda variance of the expectation value by integrating with respect to thecombined probability density function.

In one aspect, the single range measurement probability densityfunctions model an uncertainty factor including process noise, knownnetwork node location or range measurement. Preferably, the single rangemeasurement probability density functions model uncertainty factors forboth process noise and known network node location. Also preferably, theprobability density functions are Gaussian distributions.

In one embodiment, integrating with respect to the combined probabilitydensity function includes performing a lattice point integration.Alternatively, integrating with respect to the combined probabilitydensity function includes performing a Gauss-Hermite quadraturecalculation.

Another aspect of the disclosure is directed to also assessing thevariance of the expectation value to determine the validity of theposition determination for the first network node. Yet another aspect isdirected to obtaining range measurements for distances between the firstnode and the plurality of known network nodes by performing a round triptime calculation. Preferably, obtaining range measurements may alsoinclude taking at least one non-simultaneous measurement. A furtheraspect includes supplementing the position determination with positioninformation from an independent source.

This disclosure is also directed to a wireless network having positiondetermination capabilities, wherein the network includes a first networknode, a plurality network nodes having known location and aprobabilistic positioning engine configured to obtain positioninformation for the plurality of known network nodes, obtain rangemeasurements for distances between the first node and the plurality ofknown network nodes, determine a single range measurement probabilitydensity function for a position of the first network node for each ofthe range measurements, determine a combined probability densityfunction by taking the product of the single range measurementprobability density functions, and make a position determinationcomprising an expectation value corresponding to the first networknode's position and a variance of the expectation value by integratingwith respect to the combined probability density function.

Preferably, the probabilistic positioning engine is configured todetermine a single range measurement probability density function for aposition of the first network node for each of the range measurements bymodeling an uncertainty factor including process noise, known networknode location and range measurement. More preferably, the single rangemeasurement probability density function models uncertainty factors forboth process noise and known network node location. Further, suchprobability density functions are preferably Gaussian distributions.

In one aspect, the probabilistic positioning engine is configured tointegrate with respect to the combined probability density function byperforming a lattice point integration. Alternatively, the probabilisticpositioning engine is configured to integrate with respect to thecombined probability density function by performing a Gauss-Hermitequadrature calculation.

In another aspect of the disclosure, the probabilistic positioningengine also assesses the variance of the expectation value to determinethe validity of the position determination for the first network node.

A further aspect involves configuring the probabilistic positioningengine to obtain range measurements for distances between the first nodeand the plurality of known network nodes by performing a round trip timecalculation. Preferably, the probabilistic positioning engine may alsobe configured to obtain range measurements for distances between thefirst node and the plurality of known network nodes by taking at leastone non-simultaneous measurement.

Yet another aspect of the disclosure is directed to configuring theprobabilistic positioning engine to supplement the positiondetermination resulting from the integration of the combined rangemeasurement probability density functions with position information froman independent source.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features and advantages will become apparent from the followingand more particular description of the preferred embodiments of theinvention, as illustrated in the accompanying drawings, and in whichlike referenced characters generally refer to the same parts or elementsthroughout the views, and in which:

FIG. 1 is a schematic representation of a wireless network embodyingaspects of the invention;

FIG. 2 is a flow chart showing the primary steps of a probabilisticpositioning algorithm, according to the invention;

FIG. 3 is graphical representation of three single range measurementprobability density functions, according to the invention; and

FIG. 4 is graphical representation of a combined range measurementprobability density function, according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

At the outset, it is to be understood that this disclosure is notlimited to particularly exemplified materials, architectures, routines,methods or structures as such may, of course, vary. Thus, although anumber of such options, similar or equivalent to those described herein,can be used in the practice or embodiments of this disclosure, thepreferred materials and methods are described herein.

It is also to be understood that the terminology used herein is for thepurpose of describing particular embodiments of this disclosure only andis not intended to be limiting.

Some portions of the detailed descriptions which follow are presented interms of procedures, logic blocks, processing and other symbolicrepresentations of operations on data bits within a computer memory.These descriptions and representations are the means used by thoseskilled in the data processing arts to most effectively convey thesubstance of their work to others skilled in the art. In the presentapplication, a procedure, logic block, process, or the like, isconceived to be a self-consistent sequence of steps or instructionsleading to a desired result. The steps are those requiring physicalmanipulations of physical quantities. Usually, although not necessarily,these quantities take the form of electrical or magnetic signals capableof being stored, transferred, combined, compared, and otherwisemanipulated in a computer system.

It should be borne in mind, however, that all of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities. Unlessspecifically stated otherwise as apparent from the followingdiscussions, it is appreciated that throughout the present application,discussions utilizing the terms such as “accessing,” “receiving,”“sending,” “using,” “selecting,” “determining,” “normalizing,”“multiplying,” “averaging,” “monitoring,” “comparing,” “applying,”“updating,” “measuring,” “deriving” or the like, refer to the actionsand processes of a computer system, or similar electronic computingdevice, that manipulates and transforms data represented as physical(electronic) quantities within the computer system's registers andmemories into other data similarly represented as physical quantitieswithin the computer system memories or registers or other suchinformation storage, transmission or display devices.

Embodiments described herein may be discussed in the general context ofcomputer-executable instructions residing on some form ofcomputer-usable medium, such as program modules, executed by one or morecomputers or other devices. Generally, program modules include routines,programs, objects, components, data structures, etc., that performparticular tasks or implement particular abstract data types. Thefunctionality of the program modules may be combined or distributed asdesired in various embodiments.

By way of example, and not limitation, computer-usable media maycomprise computer storage media and communication media. Computerstorage media includes volatile and nonvolatile, removable andnon-removable media implemented in any method or technology for storageof information such as computer-readable instructions, data structures,program modules or other data. Computer storage media includes, but isnot limited to, random access memory (RAM), read only memory (ROM),electrically erasable programmable ROM (EEPROM), and flash memory or anyother medium that can be used to store the desired information.

As used herein, the terms “location” and “position” may be usedinterchangeably and in general refer to the physical manifestation of anetwork node with reference to a suitable coordinate system in two orthree dimensional space, depending upon the application. Accordingly,the terms “location information” and “position information” generallyrefer to any data indicating such location or position. Finally, theterms “known position” and “known location” generally refer to a networknode for which sufficient position information of adequate validity isavailable to allow its use in the position determinations of thedisclosure. The position information of a network node having a knownlocation may be derived from any suitable source, including withoutlimitation the probabilistic position determinations that are thesubject of this disclosure, any of the conventional wireless positioningtechniques described above, site surveys, manual input, and externalpositioning systems such as satellite-based navigation systems.

Further, embodiments are discussed in specific reference to wirelessnetworks. As such, this disclosure is applicable to any suitablewireless network having the necessary characteristics, includingwireless local area networks (WLAN), particularly those governed by IEEE802.11 protocols, as well as wireless fidelity (WiFi), Wibree™, ultrawideband (UWB), Long Term Evolution (LTE), Enhanced Data for GSMEvolution (EDGE), Evolution Data Optimized (EVDO), General Packet RadioService (GPRS) networks and others.

In the figures, a single block may be described as performing a functionor functions; however, in actual practice, the function or functionsperformed by that block may be performed in a single component or acrossmultiple components, and/or may be performed using hardware, usingsoftware, or using a combination of hardware and software. Also, theexemplary wireless network devices may include components other thanthose shown, including well-known components such as a processor, memoryand the like.

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one having ordinaryskill in the art to which the disclosure pertains.

Further, all publications, patents and patent applications cited herein,whether supra or infra, are hereby incorporated by reference in theirentirety.

Finally, as used in this specification and the appended claims, thesingular forms “a, “an” and “the” include plural referents unless thecontent clearly dictates otherwise.

As noted above, this disclosure is directed to systems and methods forproviding position determination in a wireless network using estimationsincorporating probability densities. The position of a network node maybe determined using range measurements to other network nodes havingknown locations. Incorporating probability functions in the estimationalgorithms may help account for the dynamic nature of wireless networks,including the relative motion of a node in the network. In so doing, thetechniques of this disclosure may reduce or avoid the need forfiltering.

Further, the use of probability functions allows the return of asolution even when the problem is underdetermined, such as whenrelatively few range measurements are available. For example, a minimumof four measurements may be required to perform a conventionallateration-based position determination yielding a solution in threedimensions. As discussed above, conventional techniques, such as thosebased on least squares estimations, may be unsuitable when only a fewmeasurements are available. In contrast, by using the estimationtechniques of this disclosure, a solution in the form of a probabilitydensity function may be obtained even when the problem isunderdetermined. Further, the solution may be more easily assessed forvalidity.

The incorporation of user dynamics in the estimation algorithm alsopermits the use of non-simultaneous measurements, so that priormeasurements may still be employed in the positioning determination tosupplement any current measurements.

Thus, the solutions to position determinations using the techniques ofthis disclosure preferably may be returned in the form of a probabilitydensity function. A probability density function may be determined foreach available range measurement. Preferably, the probability densityfunction is configured to compensate for uncertainty factors in theinformation used to make the position determination, including motion ofthe network node, errors in the location information for the knownnetwork nodes used in the range measurements, and errors in the rangemeasurements themselves. As such, the solutions may include an average,or expectation value, corresponding to the most likely position of thenetwork node together with a variance. The quality and number ofmeasurements used, together with the geometry of the nodes involved inthe determination may affect the variance. Accordingly, when thevariance is sufficiently low, the position determination may be taken asa good fix and used for any suitable purpose. Similarly, when thevariance is too high, it may be determined that the position fix is notsufficiently reliable.

These and other aspects of the disclosure are discussed in detail belowwith specific reference to exemplary embodiments.

As shown in FIG. 1, a wireless network 100 embodying aspects of thisdisclosure generally includes various nodes, such as a plurality ofaccess points 102-1, 102-2, 102-3-102-n and client device 104. Here, therange between client device 104 and access point 102-1 is shown asdistance R₁, the range between client device 104 and access point 102-2is shown as distance R₂ the range between client device 104 and accesspoint 102-3 is shown as distance R₃, and the range between client device104 and access point 102-n is shown as distance R_(n). In thisdepiction, each access point 102 is in wireless communication withclient device 104, but in typical applications, the network will besignificantly larger and may include different types of nodes, not allof which may be within range of each other.

For the purposes of this embodiment, the locations of access points 102are known to a sufficient degree of accuracy to permit their use inrange-based position determinations. The position of client device 104is unknown and an estimation of its position may be made using theprobabilistic techniques of this disclosure. As will be appreciated, theavailability of location information for the network nodes may varydepending upon circumstances, but in general, nodes having knownlocations may be used to determine the position of nodes with unknownlocations. In a different embodiment, for example, the position ofclient device 104 may be known and may be used to determine the positionof an access point 102 having an unknown location.

A position server 106 is connected to the network 100, either through awired connection, such as Ethernet, or wirelessly. In the embodimentshown, position server 106 includes a probabilistic positioning engine108 that is configured to obtain information from network 100 regardingthe characteristics of communications between access points 102 andclient device 104. Although depicted as a separate device, the functionsof position server 106 can also be implemented within one of the accesspoints 102 or client device 104, as desired.

Probabilistic positioning engine 108 also preferably maintains adatabase of position information for network 100. This includes positiondeterminations made with regard to client device 104 as well as knownpositions for relatively static nodes, such as access points 102. Asdesired, probabilistic positioning engine 108 may also be configured tocombine a current position determination with a previous positiondetermination performed by probabilistic positioning engine 108 or withan independent supplemental source of position information to refine thelocation estimation. As shown in FIG. 1, suitable sources ofsupplemental position information may include position determinationsobtained from a conventional positioning engine 110 or positioninformation from external position source 112. In one embodiment,external position source 112 may be a global navigation satellite system(GNSS) receiver associated with one of the network nodes.

Accordingly, one embodiment of the disclosure incorporates theprobabilistic positioning algorithm represented by the flowchart in FIG.2 to make a position determination regarding client device 104.Depending upon the desired application, this procedure can be adapted toeither two or three dimensions as desired.

Starting with step 202, available range measurements, distance R,regarding client device 104 are sent from network 100 to probabilisticpositioning engine 108. Depending upon the application, the rangemeasurements may be the actual distances or they may take the form oftiming measurements from which the distance may be calculated byprobabilistic positioning engine 108, for example. As discussed indetail below, a currently preferred source of range measurements is RTT,but any suitable source of range measurement may be employed. In step204, position information, location a, regarding any of the accesspoints 102 used in the range measurements is obtained from positionserver 106. Since the range measurements may be simultaneous ornon-simultaneous, each measurement has an associated time delay t,corresponding to the age of the measurement.

Next, in step 206, an uncertainty factor corresponding to process noiseis modeled using a probability distribution. Process noise reflectsrelative motion of client device 104 with respect to the known locationsof access points 102 and can be viewed as the possible potentiallocations x of client device 104 after having moved over the course oftime t from a starting location y. Preferably, a stochastic model isused, so that no knowledge regarding future motion of client device 104is assumed. In a currently preferred embodiment, motion of client device104 is modeled with Brownian dynamics, resulting in a Gaussianprobability distribution having a variance of σ_(u) ² as discussedbelow. In alternative embodiments, other suitable probabilitydistributions may be chosen as desired based on anticipated conditions.

Further, access point 102 location a may not be accurately known, eitherdue to errors in whatever technique was used to determine the locationor other circumstances. Accordingly, in step 208, an uncertainty factorregarding access point 102 location a is also preferably modeled using aprobability distribution. In a currently preferred embodiment, sucherrors are modeled with a Gaussian distribution having a variance ofσ_(a) ². In alternative embodiments, other suitable probabilitydistributions may be chosen as desired based on anticipated conditions.

In some embodiments, it may also be desirable to provide a probabilitydistribution for additional uncertainty factors, such as a probabilitydistribution that models errors in range measurements R. In suchembodiments, any suitable probability distribution may be chosen asdesired based on anticipated conditions.

Next, in step 210, an integration over the probability distributions,including those from steps 206 and 208, are taken to calculate thesingle range measurement probability density functions p(a, R, σ_(u) ²,sigma; x, t) as described below. Then, the multiple range measurementsare combined in step 212 by multiplication of the single rangemeasurement probability density functions to get the a combinedprobability density function p(x), as also described below.

Then, in step 214, an integration with respect to the combinedprobability density function is performed to calculate the expectedposition pos of the client device 104 and the variance var, whichreflects the uncertainty of the position determination and may be usedto assess the quality of the fix. In one currently preferred embodiment,the combined probability density function is integrated over a boundedvolume around access points 102 using a lattice point integration.Alternatively, the combined probability density function is integrateduse Gauss-Hermite quadrature. Each aspect is described in detail below.However, other suitable techniques for calculating the integrals may beemployed as desired.

Following calculation of the expected position and variance of clientdevice 104, the position information may be delivered to network 100 instep 216 to be used for any suitable purpose. For example, informationsent to client device 104 over network 100 may be tailored to reflectthe position of client device 104 as part of a LBS application.Alternatively, the position of client device 104 may be registered andstored by position server 106 to facilitate tracking. The position ofclient device 104 may also be reported to the user of the device fornavigational or other purposes. In one aspect, the quality of theposition determination can be assessed on the basis of the variance toidentify whether the estimation is sufficiently valid for its intendedpurpose.

As referenced above, a currently preferred model of the process noiseuncertainty factor is a Gaussian distribution that reflects Browniandynamics. As known to those of skill in the art, Brownian motion is usedto describe the random motion of particles in a media and has beensuccessfully used to model a number of real world situations. Tofacilitate the calculations, the following probability distributionmodels include isotropic diffusion. However, different probabilitydistribution models employing other diffusion characteristics may besubstituted as desired according to the intended application.

The Gaussian distribution modeling the movement of client device 104 isgiven by the probability density function:

$\begin{matrix}{{{??}\left( {{y \cdot \sigma_{u}^{2}}{t:x}} \right)} = {\frac{1}{{\sqrt{2\pi\;\sigma_{u}^{2}t}}^{3}}{{\mathbb{e}}^{\frac{{({x - y})}^{2}}{2\sigma_{u}^{2}t}}.}}} & (1)\end{matrix}$wherein the starting location of client device 104 is point y and pointx is the location of client device 104 after time t, with a variance ofσ_(u) ².

A range measurement from network 100 as described with respect to step202 is the distance R to access point 102 at the point a having a timedelay t. Accordingly, after time t, the location of the user isdescribed by the probability density

$\begin{matrix}{{p\left( {a,R,{\sigma_{u};x},t} \right)} = {{\frac{1}{{\sqrt{2\pi\;\sigma_{u}^{2}t}}^{3}}{\int_{S^{2}{({a,R})}}{{\mathbb{e}}^{- \frac{{({x - y})}^{2}}{2\sigma_{u}^{2}t}}\frac{\mathbb{d}\; y}{4\pi\; R^{2}}}}} = {\frac{1}{4\pi\; R{{{x - a}}}\sqrt{2{\pi\sigma}_{u}^{2}t}}{\left( {{\mathbb{e}}^{- \frac{{({{{{x - a}}} - R})}^{2}}{2\sigma_{u}^{2}t}} - {\mathbb{e}}^{- \frac{{({{{{x - a}}} + R})}^{2}}{2\sigma_{u}^{2}t}}} \right).}}}} & (2)\end{matrix}$

As discussed above regarding step 208, a probability density function isalso preferably used to model the uncertainty factor corresponding toerrors in the locations of access points 102. In this embodiment, aGaussian distribution is used to model these errors, again assumingisotropic diffusion, such that location a for access point 102 is themean and there is a variance of σ_(a) ². As indicated in step 210,performing another integration of equation (1) incorporating thisGaussian density yields the single range measurement probabilitydensity:

$\begin{matrix}{{p\left( {a,R,\sigma_{u},{\sigma_{a}:x},t} \right)} = {{\frac{1}{{\sqrt{2{\pi\sigma}_{a}^{2}}}^{3}}{\int_{R^{3}}{{\mathbb{e}}^{- \frac{{({y - a})}^{2}}{2\sigma_{a}^{2}}}{p\left( {y,R,{\sigma_{u};x},t} \right)}{\mathbb{d}y}}}} = {{\frac{1}{4\pi\; R{{{x - a}}}\sqrt{2\pi\; s_{a}}}\left( {{\mathbb{e}}^{- \frac{{({{{{x - a}}} - R})}^{2}}{2s_{a}}}{\mathbb{e}}^{- \frac{{({{{{x - a}}} + R})}^{2}}{2s_{a}}}} \right)} = {{{\mathbb{e}}^{{{- R^{2}}/2}s_{a}}\frac{1}{{\sqrt{2\pi\; s_{a}}}^{3}}{\mathbb{e}}^{- \frac{{({x - a})}^{2}}{2s_{a}}}\frac{\sinh\left( {R{{{{x - a}}}/s_{a}}} \right)}{R{{{{x - a}}}/s_{a}}}} = {{\mathbb{e}}^{{{- R^{2}}/2}s_{a}}{{??}\left( {a,{s_{a};x}} \right)}\frac{\sinh\left( {R{{{{x - a}}}/s_{a}}} \right)}{R{{{{x - a}}}/s_{a}}}}}}}} & (3)\end{matrix}$wherein the combined variance s_(a)=σ_(u) ²t+σ_(a) ² reflects bothprocess noise and access point location uncertainty.

Next, with respect to step 212, multiple range measurements (a, R_(a))having different ages, t_(a), may be combined as the product of eachsingle range measurement probability function, leading to the followingequation, representing the combined, non-normalized probability densitythat estimates the position of client device 104:

$\begin{matrix}{{p(x)} = {{\prod\limits_{a}{p\left( {a,R_{a},\sigma_{u},{\sigma_{a};x},t_{a}} \right)}} = {{\prod\limits_{a}{{\mathbb{e}}^{{{- R_{a}^{2}}/2}s_{a}}{{??}\left( {a,{s_{a};x}} \right)}\frac{\sinh\left( {R_{a}{{{{x - a}}}/s_{a}}} \right)}{R_{a}{{{{x - a}}}/s_{a}}}}} \propto {{{??}\left( {\left\langle a \right\rangle,{s;x}} \right)}{\prod\limits_{a}{{\mathbb{e}}^{{{- R_{a}^{2}}/2}s_{a}}\frac{\sinh\left( {R_{a}{{{{x - a}}}/s_{a}}} \right)}{R_{a}{{{{x - u}}}/s_{a}}}}}}}}} & (4)\end{matrix}$

The multiplicative constants of equation (4) may be ignored in thisapplication to facilitate computation.

Combining the variances as s_(a)=σ_(u) ²t+σ_(a) ² allows the sum of thecombined variance for each range measurement to be given as:

$\begin{matrix}{\begin{matrix}1 \\s\end{matrix} - {\sum\limits_{a}\begin{matrix}1 \\s_{a}\end{matrix}}} & (5)\end{matrix}$

Thus, the weighted mean of the access point locations with respect tothe weights 1/s_(a) may be represented as:

$\begin{matrix}{\left\langle a \right\rangle = {s{\sum\limits_{a}\frac{a}{s_{a}}}}} & (6)\end{matrix}$

Further, by setting the non-normalized expectation value to:E[ƒ]=∫ƒ(x)p(x)dx  (7)then the expectation value corresponding to the position of clientdevice 104 is:pos=^(E[x]) _(/E┌)1┐,  (8)and the variance is given by:var=^(E[x·x) ^(T) ^(]) _(/E[1]−pos·pos) _(T.)   (9)

Although the above embodiment is discussed with respect to the use ofGaussian probability distributions, other probability density functionscan be used as desired. An advantage of the techniques of thisdisclosure is the ability to tailor the specific models used to matchanticipated noise conditions and uncertainty factors by selecting theappropriate probability density function.

In the discussion above with respect to step 214, performing anintegration of the combined probability density functions results in theexpected position pos of client device 104 and the variance var of theestimation. This solution is obtained by calculating the integrals E[1],E[x] and E[x·x^(T)].

Since the probability density p(x) decays rapidly, a suitableintegration can be performed over a bounded space around the accesspoints 102. Thus, in one embodiment, the integration is a lattice pointintegration, using an array of regularly spaced points in theappropriate number of dimensions for the desired application. For athree dimensional position determination, this array may be defined as acube centered on the access points 102.

To perform the integration, a cube C a centered at access point 102 suchthat solutions to the function p(a, R, σ_(u) ², σ_(a) ²; x, t) aresufficiently small outside the cube. As will be appreciated, theappropriate size of the cube depends on the measured ranges, measurementdelays, the assumed variances, desired accuracy, computational resourcesavailable and the like. In a currently preferred embodiment, edge lengthe_(a) of the cube is given by:e _(a)=2(R _(a) +m·√{square root over (s _(a))}+c)  (10)wherein optional parameters m and c satisfy the conditions m≧3 and c≧0.A larger value for e_(a) yields higher accuracy at the expense ofincreased computational overhead.

The domain of the integrations is chosen to be the intersection of thecubes:

$\begin{matrix}{e = {\bigcap\limits_{a}e_{a}}} & (11)\end{matrix}$

When the errors of the access point location and range are large, theintersection defined by equation (11) may be empty, requiring anincrease in the parameter c. Although increasing c degrades the accuracyof the expectation value computation, the variance will still provide anindication as to whether the fix is sufficient for the intendedapplication.

Given these conditions, a solution for the integrals can be approximatedusing the finite sums:

$\begin{matrix}\begin{matrix}{{E\lbrack f\rbrack} = {\int{{f(x)}{p(x)}{\mathbb{d}x}}}} \\{\approx {\sum\limits_{X \in {\Lambda\bigcap e}}{{f(x)}{p(x)}}}}\end{matrix} & (12)\end{matrix}$

where^(Λ) is a lattice in three dimensional space. Accuracy of theestimation may be adjusted by choice of lattice spacing. In oneembodiment, a cubic lattice with lattice constant d=0.5 m may be usedand provides acceptable results in simulations. Since the choice of theintegration domain is adaptive, the computation time needed for a fixvaries.

An alternative to lattice point integration involves the use ofGauss-Hermite quadrature as suggested by the Gaussian factor in theintegrand. The approach results in the calculation of integrals of theform:

$\begin{matrix}{\begin{matrix}{{E\lbrack f\rbrack} = {\int{{f(x)}{p(x)}{\mathbb{d}x}}}} \\{= {\int{{f(x)}{g(x)}{{??}\left( {\left\langle a \right\rangle,{s;x}} \right)}{\mathbb{d}x}}}}\end{matrix}{where}} & (13) \\{{g(x)} = {\prod\limits_{a}{{\mathbb{e}}^{{{- R_{a}^{2}}/2}s_{a}}{\frac{\sinh\left( {R_{a}{{{{x - a}}}/s_{a}}} \right)}{R_{a}{{{{x - a}}}/s_{a}}}.}}}} & (14)\end{matrix}$

An affine change of variables employs the linear transformationx=√{square root over (2s)} z+<a>  (15)and using the notation:{tilde over (ƒ)}(z)=ƒ(√{square root over (2s)} z+<a>)allows the expectation value integrals to be expressed as:

$\begin{matrix}\begin{matrix}{{E\lbrack f\rbrack} \propto {\int{{f(z)}{g(z)}{\mathbb{e}}^{- z^{2}}{\mathbb{d}z}}}} \\{= {\sum\limits_{k}{{f\left( z_{k} \right)}{g\left( z_{k} \right)}{v_{k}.}}}}\end{matrix} & (17)\end{matrix}$wherein the nodes z_(k) and weights v_(k) may be selected by followingthe general Gaussian quadrature theory, for a fixed positive integer n,k=(i,j,l) with multi-indicies i,j,l=1, . . . , n, by setting nodez_(k)=(x_(i), x_(j),x_(l)) where x_(i) is the ith root of the nthLegendre polynomial P_(n) and by setting weight v_(k)=w_(i)w_(j)w_(l)where

$\begin{matrix}{w_{i} = {\frac{2}{{\left( {1 - x_{i}^{2}} \right)\left\lbrack {P_{n}^{\prime}\left( x_{i} \right)} \right\rbrack}^{2}}.}} & (18)\end{matrix}$

The values for the Gauss-Hermite nodes and weights can be calculatedbeforehand. Hence the computational cost of the algorithm is Cn³ wherethe constant C measures the effort spent on evaluating the function{tilde over (ƒ)}(z){tilde over (g)}(z) at a single node. Accordingly,the choice of n represents a trade-off between computational cost andaccuracy.

By performing step 212 using Gauss-Hermite quadrature, good accuracyaround the weighted average <a> of the access points 102 but theaccuracy is reduced as the distance from the access points increases.Accordingly, the choice between lattice point integration andGauss-Hermite quadrature may be made depending upon the expectedconditions and applications. Similarly, other suitable integrationmethods may be employed as desired.

As noted above with regard to step 202, the ranges may be obtained fromthe network 100 using RTT measurements. As known to those of skill inthe art, the RTT may be used to determine a one-way time-of-flight (TOF)for a signal transmitted between access point 102 and client device 104from a known turn-around time and by assuming symmetrical channelconditions. Multiplying TOF by the speed of light gives the distancebetween the nodes. However, any other electromagnetic range measurementcapability may also be used. As desired, the range measurementcalculations may be performed by probabilistic positioning engine 108 ormay be computed using other resources present in network 100.

RTT measurements are presently preferred due to timing advantages. Incontrast to conventional TOA and TDOA techniques, the RTT measurement isperformed using the clock of a single device, avoiding the difficult orcostly requirement of clock synchronization between the nodes beingmeasured. In general, a specific signal sent by the initiating node isautomatically returned by the destination node, allowing RTT to becalculated by the difference in time between the start of the signal atthe initiating node and the receipt of the return signal.

Necessarily, RTT techniques require a means for accurately estimatingthe delay imparted at the destination node. However, many wirelessnetwork implementations provide sufficient reproducibility to make thenecessary determinations. Accordingly, it is preferable to use a signalthat is received and returned by the destination node using operationsoccurring at the hardware level, such as the physical (PHY) layer or themedia access control (MAC) layer, rather than software operations thatare subject to a much greater degree of variability, such as thoseoccurring in the application layer. Thus, in WLAN implementations, forexample, the timing information necessary to perform RTT measurementsmay be provided by the hardware and will be available to position server106 through network 100.

To illustrate the application of the probabilistic density algorithms ofthis disclosure, an example involving three access points 102 will bediscussed with regard to determining the expectation value correspondingto the position of client device 104 and the associated variance. Tofocus on selected aspects of the algorithm, the example is artificiallysimplified to involve simultaneous range measurements applied in twodimensional space. Further, client device 104 is modeled as being staticin this example so that the process noise variances σ_(u) ² are notinvolved.

Following steps 202 and 204, the retrieved access point 102 locationsare the points a₁=(12,12), a₂=(17,17), and a₃=(20,10), and thecorresponding measured ranges are R₁=5, R₂=3, and R₃=6. Since the rangemeasurements are simultaneous, there are no time delays t. By settingthe access point location variances σ_(a) ² to 0.5, the three singlerange probability functions result in a solution graphically depicted bythe three raised rings shown in FIG. 3. The product of these singlerange probability functions is the combined probability function havingthe solution graphically represented in FIG. 4. The expectation value,corresponding to the position estimate of client device 104 ispos=(16.2, 14.5) and the variance is

${var} = {\begin{bmatrix}0.44 & {- 0.05} \\{- 0.05} & 0.33\end{bmatrix}.}$As discussed above, the small values of the variance matrix reflect thefact that the product density function shown in FIG. 4 is sharply peakedand indicates relatively little uncertainty about the positiondetermination regarding client device 104. Accordingly, by setting asuitable threshold for the values of var, the position determinationsprovided by these techniques may readily be classified as sufficientlyaccurate or not.

Described herein are presently preferred embodiments. However, oneskilled in the art that pertains to the present invention willunderstand that the principles of this disclosure can be extended easilywith appropriate modifications to other applications.

What is claimed is:
 1. A method for making a position determination fora first network node within a wireless network comprising: obtainingposition information for a plurality of network nodes, each having aknown location; obtaining range measurements for distances between thefirst network node and the plurality of network nodes having knownlocation; determining a single range measurement probability densityfunction for a position of the first network node for each of the rangemeasurements, wherein each single range measurement probability densityfunction models a process noise uncertainty factor corresponding tomovement of the first network node over time; determining a combinedprobability density function by taking a product of each of the singlerange measurement probability density functions; and determining anexpectation value corresponding to a position of the first network nodeand a variance of the expectation value by integrating with respect tothe combined probability density function.
 2. The method of claim 1,wherein determining a single range measurement probability densityfunction for a position of the first network node for each of the rangemeasurements further comprises modeling an uncertainty factor selectedfrom the group consisting of network node location error and rangemeasurement error.
 3. The method of claim 2, wherein determining asingle range measurement probability density function for a position ofthe first network node for each of the range measurements comprisesmodeling uncertainty factors for process noise and network node locationerror.
 4. The method of claim 3, wherein each of the single probabilitydensity functions modeling uncertainty factors for process noise andnetwork node location error comprise a Gaussian distribution.
 5. Themethod of claim 1, wherein integrating with respect to the combinedprobability density function comprises performing a lattice pointintegration.
 6. The method of claim 1, wherein integrating with respectto the combined probability density function comprises performing aGauss-Hermite quadrature calculation.
 7. The method of claim 1, furthercomprising assessing the variance of the expectation value to determinevalidity of the position determination for the first network node. 8.The method of claim 1, wherein obtaining range measurements fordistances between the first network node and the plurality of knownnetwork nodes comprises performing a round trip time calculation.
 9. Themethod of claim 1, wherein obtaining range measurements for distancesbetween the first network node and the plurality of network nodes havingknown location comprises taking at least one non-simultaneousmeasurement.
 10. The method of claim 1, further comprising supplementinga position determination derived from the expectation value withposition information from an independent source.
 11. A wireless networkhaving position determination capabilities comprising a first networknode, a plurality of network nodes having known location and aprobabilistic positioning engine processor configured to: obtainposition information for a plurality of network nodes, each having aknown location; obtain range measurements for distances between thefirst network node and the plurality of network nodes having a knownlocation; determine a single range measurement probability densityfunction for a position of the first network node for each of the rangemeasurements, wherein each single range measurement probability densityfunction models a process noise uncertainty factor corresponding tomovement of the first network node over time; determine a combinedprobability density function by taking a product of each of the singlerange measurement probability density functions; and calculate anexpectation value corresponding to a position of the first network nodeand a variance of the expectation value by integrating with respect tothe combined probability density function.
 12. The wireless network ofclaim 11, wherein the probabilistic positioning engine processor isfurther configured to determine a single range measurement probabilitydensity function for a position of the first network node for each ofthe range measurements by modeling an uncertainty factor selected fromthe group consisting of network node location error and rangemeasurement error.
 13. The wireless network of claim 12, wherein theprobabilistic positioning engine processor is configured to determine asingle range measurement probability density function for a position ofthe first network node for each of the range measurements by modelinguncertainty factors for process noise and network node location error.14. The wireless network of claim 13, wherein the probability densityfunctions modeling uncertainty factors for process noise network nodelocation error comprise a Gaussian distribution.
 15. The wirelessnetwork of claim 11, wherein the probabilistic positioning engineprocessor is configured to integrate with respect to the combinedprobability density function by performing a lattice point integration.16. The wireless network of claim 11, wherein the probabilisticpositioning engine processor is configured to integrate with respect tothe combined probability density function by performing a Gauss-Hermitequadrature calculation.
 17. The wireless network of claim 11, whereinthe probabilistic positioning engine processor is further configured toassess the variance of the expectation value to determine a validity ofthe position determination for the first network node.
 18. The wirelessnetwork of claim 11, wherein the probabilistic positioning engineprocessor is configured to obtain range measurements for distancesbetween the first network node and the plurality of network nodes havingknown location by performing a round trip time calculation.
 19. Thewireless network of claim 11, wherein the probabilistic positioningengine processor is configured to obtain range measurements fordistances between the first network node and the plurality of networknodes having known location by taking at least one non-simultaneousmeasurement.
 20. The wireless network of claim 11, wherein theprobabilistic positioning engine processor is further configured tosupplement a position determination derived from the expectation valuewith position information from an independent source.